English

Elliptic equations with VMO a, b$\,\in L_{d}$, and c$\,\in L_{d/2}$

Analysis of PDEs 2020-08-18 v3 Classical Analysis and ODEs

Abstract

We consider elliptic equations with operators L=aijDij+biDicL=a^{ij}D_{ij}+b^{i}D_{i}-c with aa being almost in VMO, bLdb\in L_{d} and cLqc\in L_{q}, c0c\geq0, d>qd/2d>q\geq d/2. We prove the solvability of Lu=fLpLu=f\in L_{p} in bounded C1,1C^{1,1}-domains, 1<pq1<p\leq q, and of λuLu=f\lambda u-Lu=f in the whole space for any λ>0\lambda>0. Weak uniqueness of the martingale problem associated with such operators is also obtained.

Keywords

Cite

@article{arxiv.2004.01778,
  title  = {Elliptic equations with VMO a, b$\,\in L_{d}$, and c$\,\in L_{d/2}$},
  author = {N. V. Krylov},
  journal= {arXiv preprint arXiv:2004.01778},
  year   = {2020}
}

Comments

19 p, two references added, few glitches are corrected and numerous comments by Hongjie Dong are taken into account, this version is accepted by the Trans. AMS